Calculus Examples

$lim_{x \to 0} \sqrt[5]{x^{5} - 3 x^{4} + 17} - x$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$lim_{x \to 0} \sqrt[5]{x^{5} - 3 x^{4} + 17} - lim_{x \to 0} x$

Step 2

Move the limit under the radical sign.

$\sqrt[5]{lim_{x \to 0} x^{5} - 3 x^{4} + 17} - lim_{x \to 0} x$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\sqrt[5]{lim_{x \to 0} x^{5} - lim_{x \to 0} 3 x^{4} + lim_{x \to 0} 17} - lim_{x \to 0} x$

Step 4

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

$\sqrt[5]{{(lim_{x \to 0} x)}^{5} - lim_{x \to 0} 3 x^{4} + lim_{x \to 0} 17} - lim_{x \to 0} x$

Step 5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\sqrt[5]{{(lim_{x \to 0} x)}^{5} - 3 lim_{x \to 0} x^{4} + lim_{x \to 0} 17} - lim_{x \to 0} x$

Step 6

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\sqrt[5]{{(lim_{x \to 0} x)}^{5} - 3 {(lim_{x \to 0} x)}^{4} + lim_{x \to 0} 17} - lim_{x \to 0} x$

Step 7

Evaluate the limit of $17$ which is constant as $x$ approaches $0$ .

$\sqrt[5]{{(lim_{x \to 0} x)}^{5} - 3 {(lim_{x \to 0} x)}^{4} + 17} - lim_{x \to 0} x$

Step 8

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\sqrt[5]{0^{5} - 3 {(lim_{x \to 0} x)}^{4} + 17} - lim_{x \to 0} x$

Step 8.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\sqrt[5]{0^{5} - 3 \cdot 0^{4} + 17} - lim_{x \to 0} x$

Step 8.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\sqrt[5]{0^{5} - 3 \cdot 0^{4} + 17} + 0$

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

Raising $0$ to any positive power yields $0$ .

$\sqrt[5]{0 - 3 \cdot 0^{4} + 17} + 0$

Step 9.1.2

Raising $0$ to any positive power yields $0$ .

$\sqrt[5]{0 - 3 \cdot 0 + 17} + 0$

Step 9.1.3

Multiply $- 3$ by $0$ .

$\sqrt[5]{0 + 0 + 17} + 0$

Step 9.1.4

Add $0$ and $0$ .

$\sqrt[5]{0 + 17} + 0$

Step 9.1.5

Add $0$ and $17$ .

$\sqrt[5]{17} + 0$

Step 9.2

Add $\sqrt[5]{17}$ and $0$ .

$\sqrt[5]{17}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\sqrt[5]{17}$

Decimal Form:

$1.76234034 \dots$